Mathematical Sciences: Stochastic Models for Reliability of Systems with Dependencies Among Components

提案：DMS 9503104 PI：W. J. Padgett，J. D. Lynch和S. D.达勒姆机构：南卡罗来纳州-哥伦比亚大学题目：系统可靠性的随机模型 组件之间的依赖关系 摘要： 该研究涉及复杂系统的可靠性。 主要有三个方面：（1）建立部件可靠性模型，（2）将部件可靠性和相关性结合到系统可靠性中，从而产生易处理的数据分析模型，以及（3）为这样的模型开发临界理论。 关于（1），研究了几种用于分析构件可靠性的模型，包括条件威布尔分布、逆高斯分布和Poisson-Weibull缺陷模型。 “条件威布尔”分布进行了研究，因为它适合某些纤维强度数据集以及可能是合理的，由于“删失”的考虑，在纤维制造和测试过程中。 具有有限状态马尔可夫随机强度的Poisson-Weibull缺陷模型具有混合分布（零强度）和混合危险模型（无限强度）作为极值。 该项目的这一部分的一个主要目标是一个统一的混合物理论的调查，这个缺陷模型，其中包括混合物的混合分布和混合危险的极端发展的理论，分别。 为了解决（2）和（3），研究了本质上是分层的一般模型。 层次结构包括（i）微观（或组件）级别，（ii）子系统（或“组件束”）级别，以及（iii）系统（或“束链”）级别。 人们早就知道，在许多系统中，一个组件的故障会改变施加在其余组件上的应力。 因此，将组件的依赖关系，以现实的方式到可靠性模型是非常可取的准确的结果。 在这里，组件的依赖性/相互作用被纳入模型中使用“负载共享规则”，最适用于“负载”的情况下，由于机械或物理的考虑。 一般单调的负载分配规则被认为是一种计算系统可靠度的方法。 目前的研究包括一个特殊的类，这些规则的组件负载可以使用吸收概率计算网络上的随机游走。 特别是，一些流行的负载分配规则可以减少到考虑电力网络。 能量的考虑使我们能够深入了解这些规则引起的应力集中的行为，因为组件失败。 主要目标是继续研究这些电网络规则，以及其他适用于机械载荷传递情况的网络规则，特别是复合材料，特别是研究载荷可以通过基质材料传递的“有效距离”及其与纤维断裂周围载荷通过基质传递的“无效长度”的关系。 还考虑了关键性问题。 初步工作的k-出-N系统和系统的一个小数目的组件表明，一个精确的理论可能是可获得的，当每个组件有一个威布尔故障分布使用混合变换的伽马模型，其中的变换取决于威布尔分布。 这样的模型也适用于极值近似误差的计算和使用混合分布的系统识别。 该研究涉及的模型的发展，用于描述复杂系统的依赖组件的故障。 研究人员正在研究复杂组件系统的可靠性模型，包括允许组件依赖性和相互作用的复杂材料。这些模型可直接应用于 许多当前重要的问题，包括纤维复合材料和电网的故障。 这对高可靠性复杂材料的设计和大规模生产具有重要意义。

Proposal: DMS 9503104 PIs: W. J. Padgett, J. D. Lynch and S. D. Durham Institution: University of South Carolina - Columbia Title: STOCHASTIC MODELS FOR RELIABILITY OF SYSTEMS WITH DEPENDENCIES AMONG COMPONENTS Abstract: The research involves the reliability of complex systems. There are three main thrusts: (1) modeling component reliability, (2) incorporating component reliability and dependencies into the system reliability which result in tractable data analytic models, and (3) developing a criticality theory for such models. Regarding (1), some models are studied for the analysis of component reliability, including the conditional Weibull and inverse Gaussian distributions and a Poisson-Weibull flaw model. The "conditional Weibull" distribution is studied since it fits certain fiber strength data sets well and may be justified due to "censoring" considerations in the fiber manufacturing and testing processes. The Poisson-Weibull flaw model with finite-state Markov random intensity has the mixed distribution (zero intensity) and the mixed hazard model (infinite intensity) as extremes. A major objective of this part of the project is the investigation of a unified mixture theory for this flaw model which subsumes the theory of mixtures developed for the extremes of mixed distributions and mixed hazards, respectively. To address (2) and (3), a general model is investigated which is hierarchical in nature. The hierarchy consists of (i) the micro (or component) level, (ii) the subsystem (or "bundle of components") level, and (iii) the system (or "chain of bundles") level. It has long been known that in many systems, the failure of a component changes the stress applied to the remaining components. Thus, incorporating component dependencies into a reliability model in a realistic manner is highly desirable for accurate results. Here, component dependencies/interactions are incorporated into the model by using "load-sharing rules," m ost applicable to situations where "loadings" are due to mechanical or physical considerations. General monotone load-sharing rules are considered for which a method of calculating the system reliability has been developed. The present research includes a special class of these rules where the component load can be calculated using absorption probabilities for random walks on a network. In particular, a number of the popular load-sharing rules can be reduced to the consideration of electrical networks. Energy considerations give insight into the behavior of stress concentrations induced by these rules as components fail. Major objectives are to continue the investigation of these electrical network rules, and other network rules which are appropriate for mechanical load transfer situations, especially for composite materials, and, specifically to study the "effective distance" that a load can be transferred via matrix material and its relationship to the "ineffective length" around fiber breaks for load transfer through the matrix. Criticality issues are also considered. Preliminary work for k-out-of-n systems and systems with a small number of components suggest that an exact theory may be obtainable when each component has a Weibull failure distribution using a mixed transformed gamma model, where the transformation depends on the Weibull distribution. Such a model also lends itself to calculation of extreme value approximation errors and to system identification using the mixing distribution. The research involves the development of models for describing the failure of complex systems with dependent components. The researchers are investigating reliability models for complex systems of components including complex materials which allow for component dependencies and interactions. Such models have direct application to many problems of current importance, including failure of fibrous composite materials and electrical networks. This has important imp lications for the design and large scale manufacture of complex materials where high reliability.